Liouville–Neumann series
# Liouville–Neumann series

In mathematics, the Liouville–Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory.

## Definition

The Liouville–Neumann (iterative) series is defined as

    $\phi \left(x\right)=\sum _{n=0}^{\infty }\lambda ^{n}\phi _{n}\left(x\right)$

which, provided that $\lambda$ is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind,

$f(t)=\phi (t)-\lambda \int _{a}^{b}K(t,s)\phi (s)\,ds.$

If the nth iterated kernel is defined as n−1 nested integrals of n operators K,

    $K_{n}\left(x,z\right)=\int \int \cdots \int K\left(x,y_{1}\right)K\left(y_{1},y_{2}\right)\cdots K\left(y_{n-1},z\right)dy_{1}dy_{2}\cdots dy_{n-1}$

then

    $\phi _{n}\left(x\right)=\int K_{n}\left(x,z\right)f\left(z\right)dz$

with

    $\phi _{0}\left(x\right)=f\left(x\right)~,$

so $K_0$ may be taken to be δ(x−z).

The resolvent (or solving kernel for the integral operator) is then given by a schematic analog "geometric series",

    $R\left(x,z;\lambda \right)=\sum _{n=0}^{\infty }\lambda ^{n}K_{n}\left(x,z\right).$

where $K_0$ has been taken to be δ(x−z).

The solution of the integral equation thus becomes simply

    $\phi \left(x\right)=\int R\left(x,z;\lambda \right)f\left(z\right)dz.$

Similar methods may be used to solve the Volterra equations.

